Hidden Markov Models: Missing data and multiple data streams
Ron Togunov & Marie Auger-Méthé
1 Tutorial objectives
In yesterday’s tutorial, we illustrated how to fit a basic HMM to high quality animal movement data to identify behaviours, and how to incorporate covariates on the state transition probability to identify conditions that promote different behaviours. In this tutorial, we will discuss methods to address missing data, which is very common in marine tracking data, how to integrate additional data streams (specifically, diving data), and incorporate covariates on the emission probabilities (e.g., step length depending on current speed). Specifically, we aim to address the following:
- Selecting appropriate resolution for data
- Interpolating missing locations
- Linear and statistical interpolation
- Voiding data gaps
- Path segmentation
- Multiple imputation
- Integrating diving data-streams
- Integrating covariates on emission probabilities
In this tutorial, will be using two weeks of tracking data for three narwhal provided by Dr. Marianne Marcoux.
2 Setup workspace
First, we’ll setup the workspace with required packages.
library(dplyr) # data management
library(tidyr) # data management
library(stringr) # character string management
library(data.table)# data management
library(lubridate) # date management
library(units) # physical measurement units management
library(sf) # spatial data processing
library(terra) # raster data processing
library(tmap) # mapping spatial data
library(diveMove) # calculating dive metrics
library(earthtide) # estimating tidal data
library(corrplot) # calculating Pearson's correlation matrices
library(momentuHMM)# fitting HMMs
library(conicfit) # fitting error ellipses (required by momentuHMM)
library(car) # additional operations to Applied Regression
library(mitools) # parallel processing
library(doFuture) # parallel processing
library(kableExtra)# produce visually appealing tablesMake sure to set working directory to “Day2” of the HMM workshop folder:
3 Import data and initial data processing
First, let’s import the narwhal movement data. For simplicity, we will only examine data two weeks from August, 2017 for three individuals.
The data we obtain is often quite messy with some records missing information and other records duplicated. We can filter only records with complete location data using !is.na(x) & !is.na(y). To remove duplicate records (same time, location, and location class), we will use the lag function from dplyr, which will use the value from the previous row that we can compare to the current row.
tracks <- tracks %>%
# remove missing locations
filter(!is.na(x) & !is.na(y),
# remove identical records
!(time == lag(time) & x == lag(x) & y == lag(y) & loc_class == lag(loc_class)))Next, we’ll convert the data to a spatial dataset using the sf package and plot the data. First, we define the coordinate reference system of the original data (in this case WGS84, which is defined by the EPSG code 4326). Next, we will project the data into NAD83(CSRS) UTM zone 21N (EPSG:2962), which will projected the coordinates in meter units with minimal distortion for this data set.
tracks_proj <- tracks %>%
st_as_sf(coords = c("x", "y")) %>% # converts to an sf object
st_set_crs(4326) %>% # define CRS
st_transform(2962) # reproject data to a UTMFor the main tutorial, we’ll use only the fastloc GPS data so we don’t have to deal with location error.
We lose some data, particularly near the end of the tracks. Additional processing and integration of ARGOS data is provided as a bonus at the end.
Now, we can map the data using the tmap package to visualize what it looks like. We first have to group the data by narwhal ID, and then cast the data type to LINESTRING to create paths.
# plot GPS
tracks_gps %>%
group_by(ID) %>% # group by ID
summarise(do_union = FALSE) %>%
st_cast("LINESTRING") %>% # convert data type
tm_shape() +
tm_lines(col = "ID", palette = "Dark2")
4 Selecting a time interval for the HMM
The classic HMM assumes the observation data is in discrete time and that there is no missing data in the predictor variables. There are two key decisions we must make, the temporal resolution to use, and how to address data gaps. The desired resolution depends predominantly on the biological question you are asking as different behaviours and biological processes occur at different spatial and temporal scales (e.g., seasonal migration, daily movement between foraging and resting grounds, and fine scale foraging decisions). Generally, higher resolution data is preferred as it has more information, however it is possible to have too-high of a resolution wherein information from fine-scale variability drowns out the signal from coarse-scale patterns of interest (e.g., seasonal migration). In this case, we will be linking the movement data with high resolution (75 s) dive data to look at finer-scale behaviours (on the order of a few hours). My rule of thumb, is that you want 3-50 data points per behaviour. For behaviours spanning several hours, that roughly corresponds to a desired resolution between 2 min and 60 min.
First, let’s calculate the time difference between successive records using difftime and lead (compares current row to following row). For the last record of each individual (i.e., when ID != lead(ID)), we will set to NA.
# Calculate time difference between locations
tracks_gps <- tracks_gps %>%
mutate(dt = ifelse(ID == lead(ID), # If next data row is same individual
# calculate time difference
difftime(lead(time), time, units = "mins"), NA))Let’s see what resolutions may be possible in the data by looking at the most frequent time gaps.
# Zoom in on short intervals
hist(tracks_gps$dt, 1000, main = NA, xlab = "Time difference (min)", xlim = c(0,100))
# identify the most frequent dt
tracks_gps %>%
{table(.$dt)} %>%
sort(decreasing = TRUE) %>%
head()
##
## 10 11 12 22 9 13
## 400 145 90 64 63 63We see that the most frequent time gap is 10 min, followed by 11, 12, 22, 9, and 13 min. We also see the majority of the gaps are < 60 min, however some are in excess of 600 min. Because HMMs assume there are no missing records, finer resolutions will contain more gaps that would need to be interpolated. Frequent and large data gaps can be difficult to handle, especially as the number of missing data points approaches or exceeds the existing data. Let’s examine the potential data structure at different resolutions for the different animals.
We first create a function that approximates the proportion of missing locations we would have for a given resolution.
# Make function to estimate proportion of missing location
p_na <- function(time, res) {
time <- round_date(time, paste(res,"min")) # round to nearest resolution
time <- na.omit(time[time != lead(time)]) # remove duplicate time
# calculate maximum number of locations
max_n_loc <- length(seq(time[1], tail(time, 1) + res*60,
by = as.difftime(res, units = "mins")))
n_NA <- max_n_loc - (length(time)+1)
n_NA / max_n_loc
}We can now use this function to look at the proportion of NAs we would get with 10, 20, 30, and 60 min resolution.
# summarise track dt
tracks_gps %>%
st_drop_geometry() %>%
group_by(ID) %>%
summarise(p_NA_10m = p_na(time, 10), # 10 min
p_NA_20m = p_na(time, 20), # 20 min
p_NA_30m = p_na(time, 30), # 30 min
p_NA_60m = p_na(time, 60)) %>% # 60 min
# return formatted table
kable(digits = 3, col.names = c("Narwhal ID", paste(c(10,20,30,60), "m"))) %>%
kable_styling() %>%
add_header_above(c("", "Resolution" = 4))| Narwhal ID | 10 m | 20 m | 30 m | 60 m |
|---|---|---|---|---|
| T172062 | 0.486 | 0.236 | 0.191 | 0.152 |
| T172064 | 0.502 | 0.203 | 0.120 | 0.074 |
| T172066 | 0.488 | 0.230 | 0.185 | 0.160 |
Here we see that the 10 min interval, around 50% of the locations would be missing. This is the limit that I would be comfortable at, since at finer resolutions, simulated data would outweigh real data, and may bias the results. Very large data gaps that contribute to much of the missing locations can be excluded from the analysis, therefore, for this tutorial, I will use a 10 min resolution.
5 Handling data gaps
There are several ways to deal with data gaps, and I will address four:
- Interpolation (linear and statistical)
- Voiding data gaps
- Path segmentation
- Multiple imputation
Some of these methods will not work on data with multiple individuals. To get around this, we can split the data into a list where each element of the list is the location data for each ID. We can use the R apply family functions to apply functions to each animal separately.
# convert tracks back to data.frame with xy coordinates
tracks_gps_ls <- tracks_gps %>%
# recreate x and y columns and drop sf geometry
mutate(x = st_coordinates(tracks_gps)[,"X"],
y = st_coordinates(tracks_gps)[,"Y"]) %>%
st_drop_geometry() %>%
split(.,.$ID) # split into list5.1 Approach 1: Interpolation
5.1.1 Linear interpolation
For large datasets with few and small gaps, the simplest approach is to use linear interpolation. First, let’s identify the most likely minute we have data.
# which minute has the most data
tracks_gps %>%
st_drop_geometry() %>% # convert back to data.frame
group_by(ID) %>%
summarise(minute = str_sub(minute(time), -1)) %>%
table() %>%
# create formatted table
kable() %>% kable_styling() %>%
add_header_above(c("Narwhal ID", "Minute" = 10))| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
|---|---|---|---|---|---|---|---|---|---|---|
| T172062 | 80 | 78 | 62 | 52 | 37 | 40 | 40 | 45 | 37 | 32 |
| T172064 | 77 | 68 | 71 | 67 | 32 | 32 | 40 | 32 | 31 | 38 |
| T172066 | 138 | 52 | 73 | 48 | 44 | 42 | 37 | 27 | 22 | 18 |
It looks like for all three tracks, the most amount of locations fall on 0 min (i.e., 10, 20, 30, 40, 50, or 60 min).
Next, for each track, we will create a vector of times in which to estimate locations. We will store the times for each animal as unique elements in a list.
# create full time series on which to estimate locations rounded to the nearest 10 min
tracks_gps_ls_time <- tracks_gps %>%
st_drop_geometry() %>% # convert to data.frame
group_by(ID) %>%
summarise(time = seq(round_date(first(time), "10 min"),
round_date(last(time), "10 min"),
by = 60*10)) %>%
split(.,.$ID) # split into listNow, we can interpolate the locations for each track. We will first write a wrapper function to approx that will create a data frame of approximated locations at specified times.
# function to create a data frame with approximated locations
approx_locs <- function(tracks, times){
data.frame(ID = times$ID,
time = times$time,
x = approx(tracks$time, tracks$x,
xout = times$time)$y,
y = approx(tracks$time, tracks$y,
xout = times$time)$y)
}Next, we will can apply our approx_locs function to estimate the locations for each animal using their respective regularised times. We can do this using the mapply function, which will map the function using each the following arguments as inputs to the specified function. That is, it will apply the approx_locs function iteratively to each element in the tracks_gps_ls_time and tracks_gps_ls lists.
# Interpolate the location at the times from the sequence
tracks_gps_linear <- mapply(approx_locs, tracks_gps_ls, tracks_gps_ls_time,
SIMPLIFY = F) %>%
do.call("rbind", .) # convert list of tracks back to a single data.frame
# remove row names added by rbind
rownames(tracks_gps_linear) <- NULL
# plot locations
plot(tracks_gps_linear$x, tracks_gps_linear$y, pch = 20, col = "red",
xlab = "x", ylab = "y", asp = 1)
points(st_coordinates(tracks_gps)[,"X"], st_coordinates(tracks_gps)[,"Y"], pch = 20)
Looks like it works. Let’s try fitting an HMM to this.
5.1.1.1 Fit HMM to linearly interpolated data
First, lets prepare the data using prepData and plot the data to estimate starting parameters.
prep_gps_linear <- prepData(tracks_gps_linear, type = "UTM")
# plot (just one animal)
plot(prep_gps_linear, ask = FALSE, animals = "T172062")
Note how you can see the sections where we linearly interpolated locations as a horizontal segment in step-length (e.g., around t = 100 and t = 250).
Next, we must define our model structure and initial parameters and then fit the HMM.
# define states (optional)
stateNames <- c("resident", "travel")
nbState <- length(stateNames)
# define distribution to use for each data stream
dist <- list(step = "gamma", angle = "vm")
# Setting up the starting values
mu0 <- c(50, 500) # Mean step length
sigma0 <- mu0 # SD of the step length
kappa0 <- c(0.1, 1) # Turn angle concentration parameter
# combine starting parameters
Par0 <- list(step = c(mu0, sigma0), angle = kappa0)
# Fit a 2 state HMM
set.seed(1)
HMM_gps_linear <- fitHMM(prep_gps_linear, nbState = nbState, dist = dist, Par0 = Par0, stateNames = stateNames)
# plot HMM output
plot(HMM_gps_linear, ask = FALSE, animals = "T172062")## Decoding state sequence... DONE
## Computing pseudo-residuals...
The plotted states look relatively ok, however there is quite a weird flat section in the QQ-plot of the turning angle. In data with large linearly-interpolated gaps, the model often represents the original dynamic data as one state, and defines the straight interpolated segments with a second state. This is a common problem when data gaps are frequent or large such that the information in the interpolated data outweighs the signal from the original observations. Generally, I would only use linear interpolation when data gaps are small (< 3 locations) and relatively infrequent (< 20 % of the modelled locations). In our data, some gaps are > 5 hours (30 locations) and > 50% of the modelled locations are interpolated. So we need to use another approach. The flat section in the QQ-plot around 0 emerges because the model overestimates the values just below the middle quantile (in our case, \(0^\circ\)) and just above the middle quantile. In other words, our model is not capturing the large amount of turning angles really close to \(0^\circ\) (this may be considered a good thing in our case since we know these are linearly interpolated). In addition, the model is under-estimating the prevalence of the lowest and highest turn angle quantiles.
5.1.2 Statistical interpolation
A slightly better way to interpolate locations is to fit a continuous-time correlated random walk (CTCRW) model to the data and predict the most likely locations. momentuHMM contains wrapper functions to interpolate missing locations by fitting a CTCRW to the data based on the crawl package by Devin Johnson and Josh London. There are many options to fit the CTCRW model, and a detailed tutorial for analysis with crawl is available here: https://jmlondon.github.io/crawl-workshop/crawl-practical.html. Let’s try to fit the most basic model using the wrapper function crawlWrap. In the most basic form, we only need to provide tracking data with the columns ID, time, x, and y and specify the desired temporal resolution.
# crawl can fail to fit periodically, so I recommend always setting a seed
set.seed(5)
# fit crawl
crw_gps_10 <- crawlWrap(obsData = tracks_gps, timeStep = "10 mins")
## Fitting 3 track(s) using crawl::crwMLE...
## DONE
##
## Predicting locations (and uncertainty) at 10 mins time steps for 3 track(s) using crawl::crwPredict... DONE
# view that all parameters were propperly estimated
crw_gps_10$crwFits %>%
lapply(function(x){
x[c("par","se","ci")] %>% # get estimated values
unlist() %>% # unlist
is.nan() %>% # identify values that failed to estimate
sum() == 0 # count failed estimates and ensure there are 0
})
## $T172062
## [1] TRUE
##
## $T172064
## [1] TRUE
##
## $T172066
## [1] TRUE
# plot fit track (for just one track)
plot(crw_gps_10, ask = FALSE, animals = "T172062")
Notice how the predicted tracks do not make perfectly straight lines through missing sections (particularly noticeable in T172062). Next, we will extract the predicted locations and add them to the observed data.
# filter predicted locations
tracks_gps_crw <- data.frame(crw_gps_10$crwPredict) %>%
filter(locType == "p") %>%
dplyr::select(mu.x, mu.y, time,
ID) %>%
dplyr::rename(x = "mu.x", y = "mu.y")5.1.2.1 Fit HMM to statistically interpolated data
Now, let’s try to fit the same HMM as before on this data using the same model structure and starting parameters as the HMM_gps_linear HMM.
set.seed(1)
# prep data
prep_gps_crw <- prepData(tracks_gps_crw, type = "UTM")
# Fit a 2 state HMM
HMM_gps_crw <- fitHMM(prep_gps_crw, stateNames = stateNames,
nbState = nbState, dist = dist, Par0 = Par0)
# plot outputs (for one animal)
plot(HMM_gps_crw, ask = FALSE, animals = "T172062")## Decoding state sequence... DONE
## Computing pseudo-residuals...
That’s looking much better. It looks like state 1 represents a low-speed, high tortuosity resident state, while state 2 represents higher-speed, low tortuosity travelling state. The QQ plot for turn angle, in particular, has lost the flat section around the middle quantile. However, we also see that we are still underestimating the prevalence of high turn angles, though not to the same degree as in the linearly interpolated data.
For many applications, this model may be sufficient. However, the significant proportion of interpolated locations used to fit the model is likely to affect our results. For example, the large interpolated gaps are still relatively straightened out and a very consistent speed, and may skew the definition of state 2 in particular. Simply interpreting the results with this issue in mind can be adequate when we only have few interpolated locations. Below, we will explore more formal ways to address this problem, which can be particularly useful when we have very large numbers of interpolated locations.
5.2 Approach 2: Voiding data gaps
One strategy to address large data gaps is to void the data streams (i.e., step length and turning angle) during moderate/large interpolated gaps where we expect that the estimated movement is largely independent of the observed data before or after the gap. For my own work, I have typically voided sections with \(>6\) missing locations. The maximum size of a gap to allow depends on the frequency of the missing data, frequency of locations, study species, and behaviours of interest. In this case, I will void the data streams from locations predicting in gaps \(>60\) min. First, we will identify which steps need to be voided, then we will prepare the data and void the estimated step and angle data streams. We will do this again later in the tutorial, so we will wrap it into a function called prepData_NAGaps. The function will output a momentuHMMData object that can directly be fit using fitHMM.
# define function to replace step and turn angle of large gaps with NA
NA_gaps <- function(tracks, times){
# rows where location is within a large gap
rows <- which(
rowSums(apply(times, 1, function(X, tracks){
dplyr::between(tracks,
as.POSIXct(X[1], tz = "UTC"),
as.POSIXct(X[2], tz = "UTC"))
}, tracks$time))==1)
tracks$step[rows] <- NA
tracks$angle[rows] <- NA
return(tracks)
}
# define function to identify and nullify gaps
prepData_NAGaps <- function(track_list, tracks_crw, res, max_gap, ...){
# for each ID, identify which rows have gaps >= max_gap
gaps_ls_rows <- lapply(track_list, function(x){
which(difftime(lead(x$time), x$time, units = "mins") >= max_gap)
})
# create sequence of times for each track from gaps >= 60 min
gap_ls <- mapply(FUN = function(track, gaps){
# identify start and end date of each gap
gaps_ls_srt_end <- list(start = ceiling_date(track$time[gaps], paste(res, "min")),
end = floor_date(track$time[gaps+1], paste(res, "min")))
# combine into single vector for each track
data.frame(start = gaps_ls_srt_end$start, end = gaps_ls_srt_end$end)
},
track_list, gaps_ls_rows, SIMPLIFY = F)
# prep data and list by ID
prep_tracks <- prepData(tracks_crw, ...) %>%
{split(., .$ID)}
# Interpolate the location at the times from the sequence
mapply(FUN = NA_gaps, prep_tracks, gap_ls,
SIMPLIFY = F) %>%
do.call("rbind", .) # convert list of tracks back to a single data.frame
}
prep_tracks_gps_crw_NAgaps <- prepData_NAGaps(tracks_gps_ls, tracks_gps_crw,
10, 60, type = "UTM")5.2.0.1 Fit HMM to CRW data with voided gaps
Now, let’s try to fit the same HMM as in sections 5.1.1.1 and 5.1.2.1 to this data with large gaps voided.
set.seed(1)
# Fit a 2 state HMM
HMM_gps_crw_NAgaps <- fitHMM(prep_tracks_gps_crw_NAgaps,
stateNames = stateNames, nbState = 2,
dist = dist, Par0 = Par0)
# plot outputs (for one animal)
plot(HMM_gps_crw_NAgaps, ask = FALSE, animals = "T172062")## Decoding state sequence... DONE
## Computing pseudo-residuals...
HMM_gps_crw$mle[c("step", "angle")]
## $step
## resident travel
## mean 259.1340 709.6107
## sd 154.7562 255.2652
##
## $angle
## resident travel
## mean 0.000000 0.00000
## concentration 2.520006 22.09174
HMM_gps_crw_NAgaps$mle[c("step", "angle")]
## $step
## resident travel
## mean 282.0541 754.6385
## sd 153.8700 251.0856
##
## $angle
## resident travel
## mean 0.000000 0.00000
## concentration 2.277193 18.01756Visually, the difference is subtle. However, the estimated parameters are quite different whether you account for the large gaps in data. When you void large gaps, the mean step length for both states is higher, and the turn angle concentration parameters is lower for both states (i.e., more tortuous). The fact that the parameters change for both states, suggests that the large gaps skewed the parameterisation of both states.
5.3 Approach 3: Path segmentation
Another strategy to deal with larger gaps is to segment the tracks with a new individual ID. This may be particularly appropriate for gaps where we may reasonably expect that the the underlying states are effectively independent of one another. Specifically, we may ask, over what period of time does the behaviour of the animal affect the subsequent behaviour. In this case, we may expect that the behaviour of a narwhal depends on the behaviour over the proceeding several hours, however is independent after 24 hours. We can split the tracks for gaps larger than a predetermined threshold by iterating the ID column. We will not implement this approach in this tutorial, however, path segmentation can be implemented using the following code:
gap_thresh <- 3*60 # in hours (3h for illustration)
# gaps no more than 6h per segment
# if dif.time > gap_thresh or new ID
new_ID <- (lag(tracks_gps$dt) > gap_thresh | tracks_gps$ID != lag(tracks_gps$ID)) %>%
replace_na(TRUE) %>% # replace first NA with ID = 1
cumsum() %>% # iterate ID
paste(tracks_gps$ID, ., sep = "_")
# then smaller gaps <= gap_thresh can be interpolated with crawlWrap
# create new segmented track
tracks_segmented <- tracks_gps %>%
mutate(ID = new_ID)
# identify which segments have sufficient data (here, 10 locations)
segement_ID <- tracks_segmented$ID %>%
{names(table(.))[table(.)>10]}
# filter track segments
tracks_segmented <- tracks_segmented %>%
filter(ID %in% segement_ID)
# plot
tracks_segmented %>%
group_by(ID) %>% # group by ID
summarise(do_union = FALSE) %>%
st_cast("LINESTRING") %>% # convert data type
tm_shape() +
tm_lines(col = "ID", palette = "Dark2")
## Fitting 18 track(s) using crawl::crwMLE...
## DONE
##
## Predicting locations (and uncertainty) at 10 mins time steps for 18 track(s) using crawl::crwPredict... DONEplot(crw_sgmt_10, compact = T, ask = FALSE,
animals = c("T172062_1", "T172062_2","T172062_3",
"T172062_4", "T172062_5", "T172062_6","T172062_7"))
5.4 Approach 4: Multiple imputation
The last approach to estimate missing locations and regularise the data is multiple imputation (Hooten et al. 2017; McClintock 2017). Multiple imputation works by first fitting a CTCRW model to the original data, second, drawing (i.e., simulating) a number of realisations of the position process based on the CTCRW model, third, fitting HMMs to each of the simulated realisations, and finally, pooling the estimated parameters.
momentuHMM has several functions to implement multiple imputation. The function MIfitHMMW can be used both to simulate realisations of a fitted CTCRW and fit HMMs to each one. The number of simulations is specified with nSims. We can simulate realisations without fitting HMMs by setting fit = FALSE. Here, let’s use the CTCRW model that we fit in section 5.1.2 to simulate 4 tracks using MIfitHMMW and plot them over the original track.
set.seed(1)
# simulate 4 realisations of the 10 min GPS CTCRW model
MI_sim_gps <- MIfitHMM(crw_gps_10, nSims = 4, fit = FALSE)## Drawing 4 realizations from the position process using crawl...
## DONE
## DONE# plot locations for first narwhal
# filter first ID from original data
track <- tracks_gps %>%
mutate(x = st_coordinates(tracks_gps)[,"X"],
y = st_coordinates(tracks_gps)[,"Y"]) %>%
filter(ID == "T172062")
# filter first ID for each simulation
sim_tracks <- lapply(MI_sim_gps$miData, function(x){
filter(x, ID == "T172062")})
# plot original track for first narwhal
plot(track$x, track$y, col = "red", xlab = "x", ylab = "y", asp = 1, type = 'l')
# plot each simulated track
mute <- mapply(function(data, col){
points(y~x, data = data, col = col, type = 'l')
}, data = sim_tracks, col = list("cadetblue1", "cadetblue2", "cadetblue3", "cadetblue4"), SIMPLIFY = FALSE)
Notice how in some areas the different simulations have generally good agreement in the likely location during gaps, while in others they diverge. Multiple imputation can be particularly powerful if we want to incorporate environmental variables, as spatially explicit variables can be extracted for each simulated track to sample the most likely conditions encountered by the animal.
As you can imagine, fitting multiple imputations can take quite a bit of time. We can speed up the computation by simulating/fitting each track in parallel using the ncores argument. We can identify the number of cores our PC has using parallel::detectCores().
# count number of available cores and use 60% of available cores
ncores <- parallel::detectCores()
ncores <- round(ncores*0.6)Next, let’s simulate 12 realisations of the CTCRW and fit HMMs to each in parallel.
set.seed(1)
# fit multiple imputation HMM to the 10 min GPS crawl model
HMM_gps_MI <- MIfitHMM(crw_gps_10, nSims = 12, ncores = ncores,
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## Fitting HMM with 2 states and 2 data streams
## -----------------------------------------------------------------------
##
## step ~ gamma(mean=~1, sd=~1)
## angle ~ vm(concentration=~1)
##
## Transition probability matrix formula: ~1
##
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[ ==================================================================================================== ] 100%## Calculating location 95% error ellipses... DONE
When plotting multiple imputed data, momentuHMMwill include estimated location error ellipses in the plots.
To better illustrate the effectiveness of multiple imputation in recovering states, we can randomly filter locations then compare state recovery from a single most likely imputed track to states from multiple-imputed tracks. First, let’s randomly filter 98 locations for each narwhal (in addition to the first or last locations so that the path length is the same),
set.seed(1)
# number of rows for each narwhal
tracks_gps_sub <- tracks_gps_ls %>%
lapply(function(x){
# number of rows for each narwhal
nrow <- nrow(x)
# rows to sample sorted chronologically
sub <- sort(c(1, nrow, # first, last,
sample(2:(nrow-1), 98))) # random 98 from the middle
# sample selection
x[sub,]
}) %>%
# convert back to a single data frame
do.call(rbind, .)Next, lets fit a CTCRW to the sampled data (100 locs pre narwhal).
## Fitting 3 track(s) using crawl::crwMLE...
## DONE
##
## Predicting locations (and uncertainty) at 10 mins time steps for 3 track(s) using crawl::crwPredict... DONE# check that all parameters were estimated
crw_gps_sub$crwFits %>%
lapply(function(x){
x[c("par","se","ci")] %>% # get estimated values
unlist() %>% # unlist
is.nan() %>% # identify values that failed to estimate
sum() == 0 # count failed estimates and ensure there are 0
})## $T172062
## [1] TRUE
##
## $T172064
## [1] TRUE
##
## $T172066
## [1] TRUENext, we can predict the most likely track using prepData on the CTCRW model, fit an HMM to this data using fitHMM, and calculate the proportion of decoded states that match the states predicted from the full data set.
# prep data
prep_gps_sub <- prepData(data = crw_gps_sub)
# Fit a 2 state HMM
set.seed(1)
HMM_gps_sub <- fitHMM(prep_gps_sub, nbState = 2, dist = dist, Par0 = Par0)## =======================================================================## Fitting HMM with 2 states and 2 data streams## -----------------------------------------------------------------------## step ~ gamma(mean=~1, sd=~1)## angle ~ vm(concentration=~1)##
## Transition probability matrix formula: ~1##
## Initial distribution formula: ~1## =======================================================================## DONE## [1] 0.4924709In this case, it appears that there is only a \(49\%\) overlap between the single imputation based on 300 sampled data points compared to HMM based on the 1492 data points in the original data. Given that there are only two states, \(49\%\) is about what we would expect the model to estimate by random chance at all.
As a comparison, let`s generate and fit 80 simulated tracks from the CTCRW fit to the sampled300 locations, then compare the states. With 80 simulations, the benefits of parallel processing become very apparent.
set.seed(1)
# fit MI HMM
HMM_gps_sub_MI <- MIfitHMM(crw_gps_sub, nSims = 80, ncores = ncores,
nbStates = 2, dist = dist, Par0 = Par0, fit = T)## Drawing 80 realizations from the position process using crawl...## Running simulator in parallel...## [ =================================------------------------------------------------------------------- ] 33%
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## -----------------------------------------------------------------------
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##
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[ ==================================================================================================== ] 100%## Calculating location 95% error ellipses... DONE## Decoding state sequences for each imputation...
## Decoding state probabilities for each imputation...
## Calculating location 95% error ellipses... DONE
## [1] 0.7453799Using multiple imputation, we see that the state overlap increases from \(49\%\) to \(74\%\). on your own time, try increasing the number of simulations even more to see how the predictions change.
5.4.1 Multiple imputation - preventing label switching
One issue that may come up when implementing multiple imputation is “label switching”. Label switching is when two or more HMMs converge effectively the same (i.e., all parameter estimates are the same), but the assign a different state to the set of parameters. For example, one HMM defines state 1 as slow and state 2 as fast, while a second HMM defines state 1 as fast and state 2 as slow. This can occur if the states have similar parameters, if the initial starting parameters are far from the optimum, or there likelihood surface is noisy or complex. The key issue that may arise from label switching, is when pooling the results from multiple imputations, the summarized parameters or estimated states would be incorrect. The best strategy to preventing state label switching is to use pseudo-design matrices (pseudo-DM) in tandem with the workBounds argument when repeatedly fitting the same HMM. Pseudo-DMs operate by mapping working parameters (columns) to natural parameters (rows). Each distribution parameter (e.g., step length mean and sd) for each state must have precisely one corresponding row in the DM. For example, a two state HMM, with one data stream (e.g., step length) that is defined by two parameters (e.g., mean and sd) must 4 rows. If the parameters for each state are completely independent, the pseudo DM would look as follows:
stepDM <- matrix(c(1,0,0,0,
0,1,0,0,
0,0,1,0,
0,0,0,1),
nrow = 4, ncol = 4,
byrow = TRUE)
rownames(stepDM) <- c("mu_1","mu_2","sd_1","sd_2")
colnames(stepDM) <- c("mu_1_i","mu_2_i", "sd_1_i","sd_2_i")
stepDM## mu_1_i mu_2_i sd_1_i sd_2_i
## mu_1 1 0 0 0
## mu_2 0 1 0 0
## sd_1 0 0 1 0
## sd_2 0 0 0 1Note: naming the rows and column is completely optional. Is addition, when DMs are used, the starting parameters must be specified on the working scale \((-\infty, \infty)\). If, for example, we want both states to share the same mean parameter, the DM would look as follows:
stepDM <- matrix(c(1,0,0,
1,0,0,
0,1,0,
0,0,1),
nrow = 4, ncol = 3,
byrow = TRUE)
rownames(stepDM) <- c("mu_1","mu_2","sd_1","sd_2")
colnames(stepDM) <- c("mu_1.2_i", "sd_1_i","sd_2_i")
stepDM## mu_1.2_i sd_1_i sd_2_i
## mu_1 1 0 0
## mu_2 1 0 0
## sd_1 0 1 0
## sd_2 0 0 1In this case, the model would estimate 3 parameters: mu_1.2_i, sd_1_i, and sd_2_i, where mu_1.2_i would represent the mean parameter for both state 1 and state 2.
Notably, we can constrain the natural parameters of one state to be greater or less than another state by using DMs in tandem with workBounds, which allows us to specify constraints on working parameters. For example, if we want state 2 to be faster than state 1, we can define the pseudo-DM and workBounds as follows:
# step DM
stepDM <- matrix(c(1,0,0,0,
1,1,0,0, # Notice the two 1s
0,0,1,0,
0,0,0,1),
nrow = 4, ncol = 4,
byrow = TRUE)
rownames(stepDM) <- c("mu_1","mu_2","sd_1","sd_2")
colnames(stepDM) <- c("b1","b2", "b3","b4")
# work bound
stepWB <- matrix(c(-Inf, 0, -Inf, -Inf, # lower bounds
Inf, Inf, Inf, Inf), # upper bounds
nrow = ncol(stepDM), ncol = 2,
dimnames = list(colnames(stepDM),
c("lower","upper")))
# view
stepDM; stepWB## b1 b2 b3 b4
## mu_1 1 0 0 0
## mu_2 1 1 0 0
## sd_1 0 0 1 0
## sd_2 0 0 0 1## lower upper
## b1 -Inf Inf
## b2 0 Inf
## b3 -Inf Inf
## b4 -Inf InfIn this case, \(\beta_1\) would represent the mean step length parameter for state 1, and the mean step length for state 2 would be represented by the sum of \(\beta_1\) and \(\beta_2\), where \(\beta_2\) would be transformed to be >0. Specifically, \(\mu_1^{(l)} = exp(\beta_1)\) and \(\mu_2^{(l)} = \exp(\beta_1 + \exp(\beta_2))\). Note: without specifying workBounds, \(\beta_2\) would not be transformed, and state 2 could be slower than state 1 (i.e., if \(\beta_2<0\), \(\exp(\beta_1) > \exp(\beta_1 + \beta_2)\)). For completeness, the following code integrates pseudo-DM and working constrains in an HMM fit multiple times using a multiple-imputation HMM (we will not run this).
# create named lists for DM and workBounds
DM <- list(step = stepDM)
WB <- list(step = stepWB)
# convert starting parameters for step length to the working scale
Par0_WB <- list(step = log(Par0$step),
angle = Par0$angle)
Par0_WB$step[2] <- 0
# fit MI HMM
HMM_gps_sub_MI_WB <- MIfitHMM(crw_gps_sub, nSims = 10, ncores = ncores,
nbStates = 2, dist = dist, fit = T,
Par0 = Par0_WB, DM = DM, workBounds = WB)Note: because we specify a DM for step length, we must log-transform the starting parameters to the working scale.
6 Integrating dive data
Let’s import the dive data and explore it.
dives <- read.csv("data/dives.csv") %>%
mutate(time = ymd_hms(time)) %>%
filter(month(time) == 8, day(time) > 7, day(time) <= 14)
head(dives)## ID time depth dt
## 1 T172062 2017-08-08 00:00:00 1.0 1.25
## 2 T172062 2017-08-08 00:01:15 7.5 1.25
## 3 T172062 2017-08-08 00:02:30 11.0 1.25
## 4 T172062 2017-08-08 00:03:45 5.5 1.25
## 5 T172062 2017-08-08 00:05:00 3.5 1.25
## 6 T172062 2017-08-08 00:06:15 1.0 1.25##
## 1.25 61.25 121.25 181.25 241.25 301.25 361.25
## 18547 52 16 6 1 1 1It appears as though there are relatively few gaps, however the gaps that exist are relatively long (> 60 mins). Therefore, I suggest that we should regularize the dive data and void large gaps.
# regularise data
dives <- dives %>%
group_by(ID) %>%
# regularise time series by 1.25 min
summarise(time = seq(first(time), last(time), by = 1.25*60)) %>%
# merge regularised time with original dive
left_join(dives, by = c("ID", "time"))## `summarise()` has grouped output by 'ID'. You can override using the `.groups`
## argument.Next, we have to summarise the time series into concrete data streams that can be modelled in the HMM. We can use the diveMove package to identify individual dives and calculate dive statistics. We must first convert the depth data to the TDR class for each whale. As we are working with multiple whales, we will use lapply to apply the createTDR to each whale – we will also use lapply for most of diveMove functions.
dives_ls <- split(dives, dives$ID)
dive_TDR <- lapply(dives_ls, function(data){
createTDR(time = data$time, depth = data$depth,
dtime = 1.25*60, file = "data/dives.csv")
})
# generate interactive plot
plotTDR(dive_TDR[[1]]) # note: try zooming in to part of the diveNext, we must use the calibrateDepth to calibrate the depth data. This function identifies wet and dry periods (for animals that haul out), applies a zero-offset correction (ZOC), and identifies all dives in the record and their phases. ZOC can be done using either the offset or filter method. In this case, we will assume the depth data is accurate and does not require an offset. We should also specify dive.thr, which represents the threshold depth below which an underwater phase should be considered a dive – in this case, we will set it at 8 m. There are many other optional parameters to identify dives that we will not get into in this tutorial (See vignette("diveMove") for details).
# calibrate TDR and identify dive phases
dive_TDR_calib <- lapply(X = dive_TDR, FUN = calibrateDepth,
zoc.method = "offset", offset = 0, dive.thr = 8)## Record is truncated at the beginning and at the end
## 39 phases detected## 818 dives detected## Record is truncated at the beginning and at the end
## 75 phases detected## 484 dives detected## Record is truncated at the beginning and at the end
## 41 phases detected## 628 dives detectedIn the interactive plot, try zooming into one of the dives and hover mouse over plot to preview the phases identified by `calibrateDepth’. The phases identified correspond to descent (D), descent/bottom (DB), bottom (B), bottom/ascent (BA), ascent (A), and surface (X). We can plot a specific dives as follows:
Now, we can calculate summary statistics for each dive using the function diveStats. There are many dive metrics that are estimated, and which ones to retain are species, data, and question specific. In this case, we will retain 8 from those calculated by diveStats: dive time, bottom time, maximum depth, bottom distance (measure of “wiggling while at the bottom), post-dive duration.
# calculate dive stats and add dive.id to each dive
dive_sum <- lapply(dive_TDR_calib, function(data){
mutate(diveStats(data, depth.deriv = FALSE), dive.id = row_number()) %>%
# select variables of interest
dplyr::select(dive.id, divetim, botttim, maxdep, bottdist, postdive.dur)})
# add dive.id with depth data to each depth record
dives_ls <- mapply(function(TDR, dives){
mutate(TDR, dive.id = dives@dive.activity$dive.id)
}, TDR = dives_ls, dives = dive_TDR_calib, SIMPLIFY = F)
# join TDR data with dive summary data
dives_ls <- mapply(function(TDR, dive_sum){
left_join(TDR, dive_sum, by = "dive.id")
}, TDR = dives_ls, dive_sum = dive_sum, SIMPLIFY = F)
# convert dive_ls back to a data.frame
dives_df <- do.call(rbind, dives_ls)MAM: 75 sec is actually quite long. Is that something that is recommended? Update: from the next comment, it’s because of the resolution. Please mention the resolution further up, when presenting the data, and refer back to it here.
Next, we will replace any bottom time of a valid dive (dive.id > 0) to 75 s, since at least some time must be spent at the bottom. Then, we will calculate one additional metric as a proxy for dive shape; the ratio of bottom time to dive time. Dives where \(\leq20\%\) of the time is spent at the bottom represent V-shaped dives, U-shaped dives are represented when \(>20\) and \(\leq50\%\) is spent at the bottom, and square dives are represented when \(>50\%\) of the time is spent at the bottom.
# replace NA bottom time of valid dives
dives_df$botttim[dives_df$dive.id > 0 & is.na(dives_df$botttim)] <- 75
# calculate proportion time at bottom
dives_df <- dives_df %>%
mutate(propbott = botttim/divetim)
# remove "dives" with no duration
dives_df <- dives_df %>%
filter(!(dive.id > 0 & is.na(divetim)))The next issue is that the dive data is at a different temporal resolution (75 s) than the location data (10 min). There are two options to include both data streams in the same HMM. First, we can choose to implement a hierarchical HMM, however, this is more complicated, and will be covered in tomorrow’s tutorial. The second, which we will use here, is to summarise the depth/dive data to a 10 min resolution and include them as additional data streams with step length and turning angle.
dives_sum <- dives_df %>%
# round time to same interval as location data
mutate(time = floor_date(time, "10 min")) %>%
# group rows by time interval
group_by(ID, time) %>%
# summarise data
summarise(NA_t = sum(is.na(depth))*1.25,
surf_t = sum(dive.id == 0)*1.25,
mean_dep = ifelse(NA_t == 10, NA, mean(na.rm = T, depth)),
max_dep = ifelse(NA_t == 10, NA, max(na.rm = T, depth)),
dive_t = ifelse(NA_t == 10, NA, sum(na.rm = T, divetim)),
bott_t = ifelse(dive_t < 5, NA, sum(na.rm = T, botttim)),
prop_bott = ifelse(dive_t < 5, NA, max(na.rm = T, propbott)),
max_dive_dep = ifelse(dive_t < 5, NA, max(na.rm = T, maxdep)),
bott_dist = ifelse(dive_t < 5, NA, max(na.rm = T, bottdist)),
post_dive_dur = ifelse(dive_t < 5, NA, max(na.rm = T, postdive.dur))) %>%
# remove -Inf values (typically error)
filter(!is.infinite(dive_t) & !is.infinite(bott_dist))## Warning in max(na.rm = T, bottdist): no non-missing arguments to max; returning
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## -Inf## `summarise()` has grouped output by 'ID'. You can override using the `.groups`
## argument.## # A tibble: 6 × 12
## # Groups: ID [1]
## ID time NA_t surf_t mean_dep max_dep dive_t bott_t
## <chr> <dttm> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 T172062 2017-08-08 00:10:00 0 3.75 17.3 31 1875 375
## 2 T172062 2017-08-08 00:20:00 0 6.25 7.56 20 1875 1125
## 3 T172062 2017-08-08 00:30:00 0 0 109. 123 7200 4200
## 4 T172062 2017-08-08 00:40:00 0 5 20.9 88 2100 1200
## 5 T172062 2017-08-08 00:50:00 0 5 9.94 24.5 1275 450
## 6 T172062 2017-08-08 01:00:00 0 1.25 53.2 76.5 4200 1575
## # … with 4 more variables: prop_bott <dbl>, max_dive_dep <dbl>,
## # bott_dist <dbl>, post_dive_dur <dbl>MAM: this gives a lot of warnings! WHat is going on?
Note, we used 2 different ifelse statements. First, ifelse(NA_t == 10, ...) ensured that we only calculate mean and maximum depth for periods where we have at least 1.25 min of depth data. Second, ifelse(surf_t > 5, ...) ensured that we only calculate dive metrics when at least half of the 10 min interval is spent in a dive, otherwise the animal is assumed to be at the surface and its dive metric is NA.
Any of the eight variables can be used as data streams in the HMM, however, including too many would significantly increase the number of parameters to estimate, and consequently computation time. Therefore, we must select which variables to use, which there are different several approaches: 1. Variable can be selected using expert on the species behaviour and research question. 2. We preferably want to avoid variable with a lot of missing data. 3. The data streams should exhibit variation and evidence of clustering or multi-modality that may be tied to the underlying behaviours. 4. Data streams with no variation or conversely are very noisy contain little information on underlying behaviour. 5. We want to avoid variables that are overdispersed and which would be difficult to describe using a statistical distribution. 6. Given that they are tied to autocorrelated behaviours, they too should exhibit some – but not too much – autocorrelation. 7. One of HMM’s key assumptions that the data stream are be independent of each other, therefore, we should avoid select highly co-linear data streams.
Assuming all the variables are biologically relevant, let’s look at structure in the data. I will also add a subjective score based on the results (first number after # symbol).
MAM: break down this chunk and add explanations for each bits
vars <- c("mean_dep","max_dep","dive_t","bott_t","prop_bott",
"max_dive_dep","bott_dist","post_dive_dur")
# First, missing data
dives_sum %>%
summarise(mean_dep = sum(is.na(mean_dep)), # 2
max_dep = sum(is.na(max_dep)), # 2
dive_t = sum(is.na(dive_t)), # 2
bott_t = sum(is.na(bott_t)), # 1
prop_bott = sum(is.na(prop_bott)), # 1
max_dive_dep = sum(is.na(max_dive_dep)), # 1
bott_dist = sum(is.na(bott_dist)), # 1
post_dive_dur = sum(is.na(post_dive_dur))) # 1## # A tibble: 3 × 9
## ID mean_dep max_dep dive_t bott_t prop_bott max_dive_dep bott_dist
## <chr> <int> <int> <int> <int> <int> <int> <int>
## 1 T172062 120 120 120 216 216 216 216
## 2 T172064 366 366 366 476 476 476 476
## 3 T172066 210 210 210 359 359 359 359
## # … with 1 more variable: post_dive_dur <int># Second, evidence of multi-modality, balanced variation,
# and no over-dispersion as the dive variables are zero-bound,
# applying a log transformation makes it easier to see structure
# in the distirbution
hist(log(dives_sum$mean_dep), 100) # 2 good structure, high dispersion
## Warning in logit(dives_sum$prop_bott): proportions remapped to (0.025, 0.975)
# Third, balanced autocorrelation
dives_sum_filter <- filter(dives_sum, ID == "T172062")
acf(dives_sum_filter$mean_dep, na.action = na.pass) # 1 ACF~8 some variability
# subjective score
data.frame(var = c("mean_dep", "max_dep", "dive_t",
"bott_t", "prop_bott", "max_dive_dep",
"bott_dist", "post_dive_dur"),
score = c(5, 6, 5,
1, 1, 4,
4, 1))## var score
## 1 mean_dep 5
## 2 max_dep 6
## 3 dive_t 5
## 4 bott_t 1
## 5 prop_bott 1
## 6 max_dive_dep 4
## 7 bott_dist 4
## 8 post_dive_dur 1From my subjective interpretation of these outputs, I think the five most promising variables to include are maximum depth, dive time, mean depth, maximum dive depth, and bottom distribution. Next, let’s merge the dive data streams the location data.
tracks_dives <- left_join(prep_tracks_gps_crw_NAgaps,
dives_sum[,c("ID", "time", "max_dep",
"dive_t", "mean_dep",
"max_dive_dep", "bott_dist")],
by = c("ID", "time"))Now, we must check for co-linearity between the data streams to select 1–3 to use in the HMM. We can check co-linearity using the Pearson’s correlation matrix.
# calculate and plot check Paerson's correlation matrix
tracks_dives[,c("max_dep", "dive_t", "mean_dep", "max_dive_dep",
"bott_dist", "step", "angle")] %>%
na.omit() %>%
cor() %>%
corrplot(method="number")
step and angle seem quit independent from all variables as does bott_dist. Unfortunately, there is quite high correlation between the first four dive data streams, se we will have to select which to use. dive_t and mean_dep have the lowest correlation with bott_dist and step, however max_dep had the highest subjective score. For the purposes of this tutorial, I will use max_dep and bott_dist.
MAM: HMM assumes that the response variables are conditionally independent base on the state. You may want to mention that. Some of that correlation could be explained by the state,
To get starting parameters, we can fit an HMM to each one independently. We will use the gamma distribution for both and for now will use the same starting parameters.
# identify whether there is 0 data
tracks_dives %>%
summarise(max_dep = sum(max_dep == 0, na.rm = T),
bott_dist = sum(bott_dist == 0, na.rm = T)) ## max_dep bott_dist
## 1 0 627# therefore, we need to include zero-mass parameters for bott_dist
# starting parameters (will use same ones for both for now)
mu0 <- c(130, 180) # mean
sigma0 <- c(60, 90) # sd
zm <- c(0.1, 0.1) # zero mass, where applicable
# fit dive-only HMMs
set.seed(1)
HMM_max_dep <- fitHMM(tracks_dives, nbStates = 2,
dist = list(max_dep = "gamma"),
Par0 = list(max_dep = c(mu0, sigma0)))
HMM_bott_dist <- fitHMM(tracks_dives, nbStates = 2,
dist = list(bott_dist = "gamma"),
Par0 = list(bott_dist = c(mu0, sigma0, zm)))Next, we can integrate these into one HMM together with step length and turning angle. When we specify the starting parameters, we want to think about how the states may look. For example, we might expect one resident state with slower movement, lower angular concentration, deeper dives, and greater at-depth variability. The second state may be travel, with faster movement, greater angular concentration, shallower dives, and less at-depth variability.
# prep model
stateNames = c("resident", "travel")
nbStates = length(stateNames)
dist = list(step = "gamma", angle = "vm",
max_dep = "gamma", bott_dist = "gamma")
# Starting Pars
# view Pars from previous HMMs
getPar(HMM_gps_crw_NAgaps)$Par # state 1 ~ resident, state 2 ~ travel## $step
## [1] 282.0541 754.6385 153.8700 251.0856
##
## $angle
## [1] 2.277193 18.017556## $max_dep
## [1] 30.15548 407.63055 29.06443 163.53662## $bott_dist
## [1] 24.2681154 317.3752361 25.1739909 203.2573833 0.2211351 0.5232927 # must switch parameter estimates in HMM_max_dep & HMM_bott_dist
# combine starting Pars
step0 <- getPar(HMM_gps_crw_NAgaps)$Par$step
angle0 <- getPar(HMM_gps_crw_NAgaps)$Par$angle
max_dep0 <- c(getPar(HMM_max_dep)$Par$max_dep[c(2,1)], # mu1, mu2
getPar(HMM_max_dep)$Par$max_dep[c(4,3)]) # sd1, sd2
bott_dist0 <- c(getPar(HMM_bott_dist)$Par$bott_dist[c(2,1)],# mu1,mu2
getPar(HMM_bott_dist)$Par$bott_dist[c(4,3)],# sd1,sd2
getPar(HMM_bott_dist)$Par$bott_dist[c(6,5)])# zm1,zm2
Par0 <- list(step = step0, angle = angle0, max_dep = max_dep0,
bott_dist = bott_dist0)Now, we can fit the HMM with movement and dive variables.
set.seed(1)
HMM_move_dive <- fitHMM(tracks_dives, nbStates=nbStates, stateNames=stateNames, dist=dist, Par0=Par0)Let’s see what it looks like.
## Decoding state sequence... DONE
## Computing pseudo-residuals...
The tracks look interesting. But there are some issues in the pseudo residuals.
Looking at the QQ plots the model appears to: - overestimate the number of fast movement steps - underestimating higher turning angles - underestimate shallow dives and over estimate deep dives - really wonky description of bott_dist The autocorrelation functions suggest there is remnant autocorrelation in step and max_dep that are not well described by the model. Together, this information suggests that there may be additional states that are not represented. Let’s try to add one more state with an intermediate speed, lower angle concentration, and shallow dives.
# define states
stateNames <- c("resident", "travel", "search")
nbStates <- length(stateNames)
# Starting Pars
# get Pars from last HMM_move_dive HMM
Pars <- getPar(HMM_move_dive)$Par
# combine starting Pars
step0 <- c(Pars$step[c(1,2)], mean(Pars$step[c(1,2)]), # mu
Pars$step[c(3,4)], mean(Pars$step[c(3,4)])) # sd
angle0 <- c(Pars$angle, 3)
max_dep0 <- c(Pars$max_dep[c(1,2)], 25, # mu
Pars$max_dep[c(3,4)], 10) # sd
bott_dist0 <- c(Pars$bott_dist[c(1,2)], # mu
mean(Pars$bott_dist[c(1,2)]),
Pars$bott_dist[c(3,4)], # sd
mean(Pars$bott_dist[c(3,4)]),
Pars$bott_dist[c(5,6)], # zm
mean(Pars$bott_dist[c(5,6)]))
Par0 <- list(step = step0, angle = angle0, max_dep = max_dep0,
bott_dist = bott_dist0)
# fit 3-state HMM
set.seed(1)
HMM_move_dive_3s <- fitHMM(tracks_dives, nbStates=nbStates, stateNames=stateNames, dist=dist, Par0=Par0)## Computing pseudo-residuals...
## Decoding state sequence... DONE
Interestingly, the step and angle QQ-plots were not improved much, though the step ACF was improved. However, compared to the 2-state model, there was a drastic improvement in QQ-plot and ACF the max_dep data stream and marginal improvement in the bott_dist data stream. The newly described state appears to have very low step length, really wide turning angle, minimal diving, and minimal at-depth activity, which may actually be indicative of resting. The “resident” state has intermediate speed and turning angle, but very deep dives and high at-depth activity, suggesting it may represent foraging. Let’s try to fit one more 4-state HMM, to try and address the remaining residuals: intermediate speed, lower angle concentration, and more intermediate dives, which may represent searching behaviour.
# define states
stateNames <- c("forage", "travel", "rest", "search")
nbStates <- length(stateNames)
# Starting Pars
# get Pars from last HMM_move_dive HMM
Pars <- getPar(HMM_move_dive_3s)$Par
# combine starting Pars
step0 <- c(Pars$step[c(1:3)], mean(Pars$step[3]), # mu
Pars$step[c(4:6)], mean(Pars$step[6])) # sd
angle0 <- c(Pars$angle, Pars$angle[3])
max_dep0 <- c(Pars$max_dep[c(1:2)], Pars$max_dep[3]/2, # mu
Pars$max_dep[3],
Pars$max_dep[c(4:5)], Pars$max_dep[6]/2, # sd
Pars$max_dep[6])
bott_dist0 <- c(Pars$bott_dist[c(1:3)], Pars$bott_dist[3], # mu
Pars$bott_dist[c(4:6)], Pars$bott_dist[6], # sd
Pars$bott_dist[c(7:9)], Pars$bott_dist[9]) # zm
Par0 <- list(step = step0, angle = angle0, max_dep = max_dep0,
bott_dist = bott_dist0)
# fit 4-state HM
set.seed(1)
HMM_move_dive_4s <- fitHMM(tracks_dives, nbStates=nbStates,
stateNames=stateNames, dist=dist, Par0=Par0)## =======================================================================## Fitting HMM with 4 states and 4 data streams## -----------------------------------------------------------------------## step ~ gamma(mean=~1, sd=~1)## angle ~ vm(concentration=~1)## max_dep ~ gamma(mean=~1, sd=~1)## bott_dist ~ gamma(mean=~1, sd=~1, zeromass=~1)##
## Transition probability matrix formula: ~1##
## Initial distribution formula: ~1## =======================================================================## DONE## Computing pseudo-residuals...
## Decoding state sequence... DONE
The QQ-plot for step is quite a bit improved, as is for bott_dist. The ACF for step is also quite improved. At this point, I’m not sure the model can be significantly improved with the existing data. Although HMMs with different number of states generally shouldn’t be compared, since AIC generally favours more states, it is a good sanity check.
MAM: maybe reference Pohle for some of these statements
## Model AIC
## 1 HMM_move_dive_4s 70947.64
## 2 HMM_move_dive_3s 71847.07
## 3 HMM_move_dive 73358.897 Covariates on emission probabilities
Thus far, we discussed various methods to regularise your data if you have missing locations and calculating and integrating dive metrics into HMMs. Further, in yesterday’s tutorial we discussed incorporating covariates on the state transition probability. momentuHMM also provides the option to include covariates on the emission probability, that is, the distribution parameters (e.g., step length mean or SD) could depend on covariates. For example, we may predict that the apparent swim speed (e.g., mean step length) may depend on ocean current speed. This section will show how to integrate covariates into the emission probabilities for mean step length and mean turning angle.
7.1 Covariates on step length emission probability
This section will illustrate how to model narwhal step length as a function of tidal currents. The narwhals in this tutorial primarily occupy the Eclipse sound system, where incoming tides lead to tidal currents into the fjords, and outgoing tides lead to tidal currents out of the fjords. Unfortunately, there is no high resolution measurement data or modelled data of tidal currents in this system. However, there are general models based on lunar and solar cycles and broad-scale coast lines that we can use for the purposes of this tutorial. Specifically, we will use the calc_earthtide function from the earthtide package to estimate horizontal displacement. This model does not take into account fine-scale coastlines, so the specific direction of tides cannot be used, but general patterns of stationary currents or tide-driven currents should be somewhat applicable. For this section, we will use the CRW data with voided gaps. First, we must project the data into lat long (WGS84), which is the required CRS for earthtide.
# re-project crw data to WGS84
prep_tracks_gps_crw_NAgaps_wgs84 <-
prep_tracks_gps_crw_NAgaps %>%
as.data.frame() %>% # convert from momentuHMM object to data frame
st_as_sf(coords = c("x", "y")) %>% # converts to an sf object
st_set_crs(2962) %>% # specify projection of daat
st_transform(4326) # re-project data to a lat long
# define x and y colums and remove sf geometry
prep_tracks_gps_crw_NAgaps_wgs84 <- prep_tracks_gps_crw_NAgaps_wgs84 %>%
mutate(x = st_coordinates(prep_tracks_gps_crw_NAgaps_wgs84)[,"X"],
y = st_coordinates(prep_tracks_gps_crw_NAgaps_wgs84)[,"Y"]) %>%
st_drop_geometry()At the small scale we are using, there is not much spatial variation in the tides, so for faster processing, we can just estimate tidal currents at each narwhal time step using the coordinates for the centre of Eclipse sound (\(79.7^\circ\) W, \(72.6^\circ\)N).
prep_tracks_gps_crw_NAgaps_wgs84 <- prep_tracks_gps_crw_NAgaps_wgs84 %>%
group_by(ID) %>% # group by narwhal ID for calc_earthtide
mutate(htide = calc_earthtide(utc = time,
do_predict = TRUE,
method = 'horizontal_displacement',
latitude = 72.6,
longitude = -79.7)[,2])
# invert the horizontal_displacement so values are strictly positive
# (I'm not sure what the units for this variable are)
prep_tracks_gps_crw_NAgaps$htide <-
-prep_tracks_gps_crw_NAgaps_wgs84$htide + 10
# plot tide data as afunction of time
plot(htide ~ time, prep_tracks_gps_crw_NAgaps, type = 'l')
If narwhal movement occurred at a larger scale (e.g., if we studied a longer period), and we wanted to estimate tidal currents independently for each location, we could use the following code (not run):
prep_tracks_gps_crw_NAgaps_wgs84 <- prep_tracks_gps_crw_NAgaps_wgs84 %>%
rowwise() %>% # must apply calc_earthtide independently to each row
mutate(htide = calc_earthtide(utc = time,
method = 'horizontal_displacement',
latitude = y, # use x,y from each row
longitude = x)[,2]) %>%
ungroup() # ungroup data so future analysis is not conducted rowwiseNext, we can define the model structure including the design matrix, which can be used to specify effects of covariates. Note: when specifying a design matrix with DM, starting parameters (i.e., Par0) must be specified on the working scale (i.e., within \((-\infty, \infty)\)), therefore, we apply a log transformation for the step length mean and SD parameters.
set.seed(1)
# define state names (optional)
stateNames <- c("resident", "travel")
# define number of states
nbState <- length(stateNames)
# define distributions for each data stream
dist <- list(step = "gamma", angle = "vm")
# define step formula and design matrix
stepDM <- list(mean = ~ htide, sd = ~ 1)
DM <- list(step = stepDM)
# define starting values
mu0 <- log(c(200, 1, # state 1 step mean: int + slope (working scale)
500, 1)) # state 2 step mean: int + slope (working scale)
sigma0 <- log(c(150, 300)) # SD of the step length (working scale)
kappa0 <- c(0.1, 1) # Turning angle concentration parameter (kappa > 0)
Par0 <- list(step = c(mu0, sigma0), angle = kappa0)Now we can fit the HMM, plot the output, and view the estimated parameters with confidence intervals. Note, when the model has covariates, you can include plotCI = T to include the 95% confidence intervals on the estimated value.
# Fit a 2 state HMM
HMM_StepTides <- fitHMM(prep_tracks_gps_crw_NAgaps, nbState = nbState,
dist = dist, Par0 = Par0, DM = DM,
stateNames = stateNames)
# plot
plot(HMM_StepTides, ask = F, plotCI = T, plotTracks = F)## Decoding state sequence... DONE
# show parameter estimates and confidence intervals
# get estimate and CI
HMM_StepTides$CIbeta$step[c("est", "lower", "upper")] %>%
lapply(as.vector) %>% # convert est, lower, and upper to vectors
do.call(cbind, .) %>% # cbind est, lower, and upper into a data.frame
# add column for beta names
data.frame(Parameter = colnames(HMM_StepTides$mle$step), .) %>%
kable(digits = 3) # output table with 3 decimal places on values| Parameter | est | lower | upper |
|---|---|---|---|
| mean_1:(Intercept) | 5.609 | 5.544 | 5.675 |
| mean_1:htide | 0.001 | -0.001 | 0.003 |
| mean_2:(Intercept) | 6.558 | 6.512 | 6.605 |
| mean_2:htide | 0.003 | 0.001 | 0.004 |
| sd_1:(Intercept) | 5.036 | 4.978 | 5.094 |
| sd_2:(Intercept) | 5.521 | 5.474 | 5.568 |
Based on the first plot and the negative and positive confidence intervals for mean_1:htide, it does not appear there is a relationship between tidal currents and speed for the resident state. However, there may be a positive relationship with speed for the travelling state. These results align with the prediction that narwhal movement is random relative to tides during the resident state and that movement tends to be with tidal current during the travelling state to save energy.
momentuHMM has special functions that allow us to specify state-specific effects of covariates, namely: state, toState, and betaCol. Using state2(htide) in the DM for step length, we specify that tidal currents only affect the travelling state (state 2) as follows:
set.seed(1)
# define starting values
mu0 <- log(c(200, 500, 1)) # Mean step length (working scale)
sigma0 <- log(c(150, 300)) # Sd of the step length (working scale)
kappa0 <- c(0.1, 1) # Turning angle concentration parameter (kappa > 0)
Par0 <- list(step = c(mu0, sigma0), angle = kappa0)
# define formula.
# state 2 mean is a function of htide
stepDM <- list(mean = ~ state2(htide), sd = ~ 1)
DM <- list(step = stepDM)
# Fit HMM
HMM_S2StepTides <- fitHMM(prep_tracks_gps_crw_NAgaps, nbState = nbState,
dist = dist, Par0 = Par0, DM = DM,
stateNames = stateNames)
# plot
plot(HMM_S2StepTides, ask = FALSE, plotCI = T, plotTracks = F)## Decoding state sequence... DONE
# show parameter estimates and confidence intervals
# get estimate and CI
HMM_S2StepTides$CIbeta$step[c("est", "lower", "upper")] %>%
lapply(as.vector) %>% # convert to vectors
do.call(cbind, .) %>% # cbind into data.frame
# add beta names
data.frame(names = colnames(HMM_S2StepTides$mle$step), .) %>%
kable(digits = 3) # output table| names | est | lower | upper |
|---|---|---|---|
| mean_1:(Intercept) | 5.644 | 5.605 | 5.684 |
| mean_2:(Intercept) | 6.566 | 6.521 | 6.611 |
| mean_2:htide | 0.003 | 0.001 | 0.004 |
| sd_1:(Intercept) | 5.039 | 4.982 | 5.097 |
| sd_2:(Intercept) | 5.521 | 5.474 | 5.568 |
Here, we see that tidal effects do appear to affect step length in state 2. As the observation data is the same between this model and the model in section 5.2, we can use AIC to compare models with and without tidal data.
AIC(HMM_gps_crw_NAgaps) # without tide data
## [1] 35787.62
AIC(HMM_StepTides) # tidal effects on state 1 and state 2 step length
## [1] 35779.65
AIC(HMM_S2StepTides) # tidal effects on state 2 step length
## [1] 35779.41In this case, the original model without tidal effects had the best AIC, suggesting that including tides did not explain enough additional variation to warrant the additional parameter.
7.1.1 Emission probability formula using pseudo-design matrix
For me, using the special functions (i.e., state and toState, and betaCol) can get confusing with more complicated models. Instead, I prefer to define pseudo-DMs explicitly so I know exactly how parameters are defined. The previous HMM can be identically defined using the following pseudo-DM (not run):
# define pseudo-design matrix
# b1 b2 b3 b4 b5
stepDM <- matrix(c(1, 0, 0, 0, 0, # mean 1 (intercept-only)
0, 1, "htide",0, 0, # mean 2 (int + slope*htide)
0, 0, 0, 1, 0, # sd 1 (intercept-only)
0, 0, 0, 0, 1),# sd 2 (intercept-only)
byrow = T, nrow = nbState*2,
dimnames = list(c("mean_1","mean_2","sd_1","sd_2"),
c("mean_1", "mean_2:int",
"mean_2:htide", "sd_1", "sd_2")))
DM <- list(step = stepDM)
# fit HMM as in previous code chunkNote: adding dimension names is optional, but can help with interpretation. Remember, each row of the DM corresponds with the state-specific distribution parameters and each column represents the coefficients estimated by the model. A basic HMM without any covariates is effectively using a diagonal matrix of 1s. Multiple values in one row are multiplied by their respective beta coefficients, then added together, and then transformed using the link function (in this case, log for mean and SD). Mathematically, the above DM can be written as follows: \[\begin{equation} \label{eq:sl}
\begin{aligned}
\log\mu_{S,t}^{(l)} &=
\begin{cases}
\begin{aligned}
& \beta_{1} \\
& \beta_{2} + \beta_{3}r^{(\mathrm{htide})}_t \\
\end{aligned}
\end{cases}
\quad
\begin{aligned}
& \mathrm{if} \: S = \mathrm{resident}\\
& \mathrm{if} \: S = \mathrm{travel}\\
\end{aligned} \\
\log\sigma_{S,t}^{(l)} &=
\begin{cases}
\begin{aligned}
& \beta_{4} \\
& \beta_{5} \\
\end{aligned}
\end{cases}
\quad
\begin{aligned}
& \mathrm{if} \: S = \mathrm{resident}\\
& \mathrm{if} \: S = \mathrm{travel},\\
\end{aligned}
\end{aligned}
\end{equation}\]
where \(\mu_S^{(l)}\) is the state-specific mean of step length, \(\sigma_{S}^{(l)}\) is the state-specific SD of step length, \(\beta_i \in (-\infty, \infty)\) are the coefficients estimated by the model, and \(r^{(htide)}_t\) is the rate of horizontal tide movement at time \(t\).
7.2 Covariates on turning angle emission probability
It is possible to include covariates on any data stream including turning angle. Thus far, we have been assuming that all movement tends to have directional persistence such that the mean turning angle in every state is \(\mu_S^{(\phi)} = 0\) and the angular concentration is state specific. If we want to include covariates on \(\kappa_S^{(\phi)}\), we can do so as in the previous section. To include covariates on the mean turning angle \(\mu_S^{(\phi)}\), we must also include the circularAngleMean argument in fitHMM and include additional starting parameters for angle (thus far we have only been specifying starting parameters for \(\kappa_S^{(\phi)}\)).
When no covariates are included on the mean turning angle, we are effectively describing the movement as a correlated random walk (CRW), wherein the only factor that determines the direction an animal moves is directional persistence (i.e., tending to move forward in the same direction). However, most behaviours exhibit some directional bias relative to the external stimuli. For example, when travelling, narwhals may orient themselves in the same direction as currents to save energy, if killer whales are present on a large scale (e.g., within 100 km) they may orient themselves toward shore as suggested by Breed et al. (2017), or if killer whales are nearby, narwhals may orient their movement directly away from killer whales. Movement where orientation is a combination of correlated directional persistence and bias relative to external stimuli is known as a biased correlated random walk (BCRW). Many interesting hypotheses can be tested by including covariates on the mean turning angle.
In this section we will examine a basic BCRW where a narwhal moves parallel to the estimated direction of tidal currents, then a more complex example where they travel parallel to the nearest coastline.
7.2.1 BCRW - bias toward or away from target
First will use the earthtide package to estimate the direction of tides; specifically, the eastward (e_w_displacement) and northward (n_s_displacement) components of tidal currents. As in the previous section, we must first project the data into WGS84:
tracks_gps_crw_wgs84 <- tracks_gps_crw %>%
st_as_sf(coords = c("x", "y")) %>% # converts to an sf object
st_set_crs(2962) %>% # specify projection of daat
st_transform(4326) As with the previous section, we will only calculate the direction in the centre of Eclipse Sound (\(79.7^\circ\) W, \(72.6^\circ\)N), as this package does not have the spatial resolution to reflect variability across our system.
tracks_gps_crw_wgs84 <- tracks_gps_crw_wgs84 %>%
group_by(ID) %>% # group by narwhal ID for calc_earthtide
mutate(tide_u = calc_earthtide(utc = time,
method = "e_w_displacement",
latitude = 72.6,
longitude = -79.7)[,2],
tide_v = calc_earthtide(utc = time,
method = "n_s_displacement",
latitude = 72.6)[,2]) %>%
ungroup()
# convert back to EPSG2962
tracks_gps_crw_wgs84 <- tracks_gps_crw_wgs84 %>%
st_transform(2962) %>% # re-project data to a lat long
# define x and y and drop geometry
{mutate(., x = st_coordinates(.)[,"X"],
y = st_coordinates(.)[,"Y"])} %>%
st_drop_geometry()Next, we can calculate the tide direction using the atan2 function. This calculates the direction as radians counter-clockwise from the eastward bearing.
Now, when we prep the data for momuntuHMM, we must specify that tide_dir is an angular covariate. Because turning angle is calculated as the angle between the step at time \(t\) relative to the step at time \(t-1\), for the angular covariate, momentuHMM will calculate the angle between the animal’s step at time \(t-1\) relative to the tides at time \(t\).
# prep data
prep_gps_crw_wgs84 <- tracks_gps_crw_wgs84 %>%
as.data.frame() %>%
prepData(type = "UTM", angleCovs = "tide_dir")As the location data is the same as the previous section, we can pull its step and angle to void large data gaps.
Before we fit the HMM, we can define the DM for turning angle. Here, I will specify the DM explicitly using pseudo-DMs.
# define turning angle pseudo-DM
# b1 b2 b3
angleDM <- matrix(c(0, 0, 0,# mean 1
"tide_dir",0, 0,# mean 2
0, 1, 0, # k 1
0, 0, 1),# k 2
byrow = T, nrow = nbState*2,
dimnames = list(c("mean_1","mean_2","k_1","k_2"),
c("mean_2:tide","k_1","k_2")))
DM <- list(angle = angleDM)In the above DM, we are effectively fixing the mean turning angle for state 1 to \(0^\circ\) and stating that the mean direction of state 2 (travel) depends on the tide direction.
Now, we will define the starting parameters for the model and fit the HMM. We must add two arguments to fitHMM. First, we must specify for which angular data streams we want to estimate the mean turning angle using estAngleMean. Second, when we are including covariates on the turning angle mean, we should also set circularAngleMean to TRUE for the turn angle. circularAngleMean will fit a special link function based on Rivest et al. (2016):
where \(\psi_t \in (-\pi,\pi]\) is the observed angle of stimulus at \(t\) relative to the movement bearing at \(t-1\) and \(\alpha_{1} \in (-\infty, \infty)\) is the bias coefficient in the direction \(\psi_t\). In this model, when \(\alpha_{1} = 0\), the model simplifies to a basic CRW with a turning angle mean centred at 0. However, as \(\alpha_{1}\) increases (or becomes increasingly negative), the turning angle becomes increasingly governed by the stimulus (i.e., toward or away from the direction of stimulus) and less by directional persistence.
set.seed(1)
# define starting values
mu0 <- c(200, 500) # Mean step length
sigma0 <- c(150, 300) # SD of the step length
angle0 <- 1 # mean 2 TA
kappa0 <- c(0.1, 1) # TA concentration parameter (kappa)
Par0 <- list(step = c(mu0, sigma0), angle = c(angle0,kappa0))
# Fit HMM
HMM_S2AngleTides <- fitHMM(prep_gps_crw_wgs84, nbState = nbState,
dist = dist, Par0 = Par0, DM = DM,
stateNames = stateNames,
estAngleMean = list(angle = T),
circularAngleMean = list(angle = T))
# plot
plot(HMM_S2AngleTides, ask = FALSE, plotCI = T, plotTracks = F)## Decoding state sequence... DONE
# show parameter estimates and confidence intervals
# get estimate and CI
HMM_S2AngleTides$CIbeta$angle[c("est", "lower", "upper")] %>%
lapply(as.vector) %>% # convert to vectors
do.call(cbind, .) %>% # cbind into data.frame
# add beta names
data.frame(names = colnames(HMM_S2AngleTides$mle$angle), .) %>%
kable(digits = 3) # output table| names | est | lower | upper |
|---|---|---|---|
| mean_2:tide | 0.007 | -0.012 | 0.026 |
| k_1 | 0.823 | 0.749 | 0.896 |
| k_2 | 2.889 | 2.786 | 2.992 |
It is clear from the second plot, that the tidal direction estimated using the earthtide package does not explain much of observed turning angle given by the confidence intervals for mean_2:tide and the second plot. In the second figure, the x-axis represents the angle of tide relative to the previous step and the y-axis is the estimated turning angle. We see that when the tide angle is \(0^\circ\), the expected mean turning angle is also \(0^\circ\) (i.e., the tide is in the same direction as directional persistence and no turn is expected). Moving to the right, as the tide angle increases (i.e., rotates towards the left), the turning angle also increases (i.e., also turns towards the left). Conversely, as the tide angle decreases (i.e., right relative to the narwhal), the narwhal’s turning angle also shifts to the right (negative values). However, throughout the plot, the estimated turning angle always encompasses \(0^\circ\), suggesting there is not a strong relationship. This result makes sense as the earthtide package does not accurately represent tides in the complex coastline of Eclipse Sound, where waterways are angled in various directions (e.g., NBI, MI, TaS, and PI). 
7.2.2 BCRW - bias parallel or perpendicular to target
Directional bias can be directly toward or away from a stimulus (e.g., resource patch, or predator location) or it may be at some angle relative to the stimulus (e.g., bird flight \(45^\circ\) relative to wind to maximise ground speed, polar bear crosswind movement during olfactory search, or sea turtle navigation relative to magnetic fields). Movement with an angled bias is known as menotaxis and was the focus of my PhD and is described Togunov et al. (2021) and Togunov et al. (2022).
We may hypothesise that when travelling, narwhal’s movement is parallel to the coastline. In the context of a BCRW, we can model this by defining turning angle as a function of the direction of the nearest coastline. We can obtain the direction of the nearest coastline using the direction function from the terra package, by definition, this angle is perpendicular to the direction of the coastline.
7.2.2.1 Calculate angle of coastline
To extract the direction to the nearest coastline, we will first import a vector shapefile, rasterise it, calculate the direction, then transform the units from clockwise from cardinal North (output by terra) to counter-clockwise from cardinal East (required by momentuHMM).
First, we will import the land shapefile using the vect function from terra
# import land polygon shapefile (project to match track if necessary)
land <- vect("data/land.shp")
plot(land)
To rasterise the land data, we must define a raster grid that would adequately encompass the narwhal tracks. This entails calculating the range of the narwhal x and y coordinates, then defining a raster with extent beyond this limit, selecting a resolution (in this case, 100 m), and a CRS (in this case, EPSG:2962). Then we can use rasterise the land polygon to the defined raster grid.
# determine range of narwhal location data
range(tracks_gps_crw$x)
## [1] -297006.3 -180506.7
range(tracks_gps_crw$y)
## [1] 8149763 8209172
# define raster grid using extent beyond narwhals, resolution, & CRS
r <- rast(ext(-320000, -150000, 8140000, 8250000),
resolution = 100, crs = "epsg:2962")
# rasterise land layer to defined raster
land_rast <- terra::rasterize(land, r)
# plot(land_rast)Third, we can calculate the direction of each water cell to the nearest land cell using the direction function. However, this will calculate direction as radians clockwise from North, while momentuHMM requires angular covariates to be in radians counter-clockwise from East. Therefore, we will also need to inverse and rotate the estimated direction to land.
# calculate direction of each water cell to nearest land cell
land_dir <- direction(land_rast)
plot(land_dir)
# invert and rotate clockwise by pi/2
land_dir <- -1*land_dir+pi/2
# use ifelse for SpatRasters to add 2*pi to cells less than -pi
land_dir <- ifel(land_dir <= -pi, land_dir+2*pi, land_dir)
plot(land_dir)
Lastly, we can extract the land direction for the narwhal data (must be a spatVector object).
# convert track to spatVector
tracks_gps_crw_vect <- tracks_gps_crw_wgs84 %>%
as.data.frame() %>%
vect(geom = c("x", "y"), crs = "epsg:2962")
# extract dir to nearest land cell
tracks_gps_crw_wgs84$land_dir <-
extract(land_dir, tracks_gps_crw_vect)[,"layer"]Now, we can prepare the data for momentuHMMand specify land_dir as the angular covariate.
# prep data
prep_gps_crw_wgs84 <- tracks_gps_crw_wgs84 %>%
as.data.frame() %>%
prepData(type = "UTM", angleCovs = "land_dir") ## Warning in prepData.default(., type = "UTM", angleCovs = "land_dir"): There
## are 27 missing covariate values. Each will be replaced by the closest available
## value.Note: the warning message may be because some locations were so close to land in reality, they fell on a cell defined as land and where there is no applicable “direction to nearest land cell”. It may also occur if the interpolated locations occur on land. In cases with missing covariate data, momentuHMM will replace the NA with the closest available value, which should be good enough for our purposes.
As the location data is the same as the previous section, we can pull its step and angle to void large data gaps.
7.2.2.2 Define and fit model
Our hypothesis is that when travelling, narwhals turning angle is biased to be parallel to the coastline, or \(\pm90^/circ\) relative to the direction to the nearest coastline. The main challenge in implementing this model is that there are two ways to move parallel relative to the coastline: up the coastline and down the coastline. For a given angular covariate, momentuHMM allows us to model angular bias toward/away one angle (i.e., either \(+90^/circ\) or \(-90^/circ\)). To include two possible directions, we have to use a trick where we divide the one behaviour into two states with their own respective biases. In this case, we will divide the travelling state into travelling left (travel_L) and travelling right (travel_R) relative to the direction to the nearest coastline.
# define model
stateNames <- c("resident", "travel_L", "travel_R")
nbState <- length(stateNames)
# define distributions for each data stream
dist <- list(step = "gamma", angle = "vm")Next, we must define the pseudo-DM, where travel_L is biased toward land_dir + pi/2 and travel_R is biased toward land_dir - pi/2. In formula notation, when we want to execute a mathematical operation as written, we must use the R function I (see help("I") for details). Second, because travel_L and travel_R represent the same biological behaviour that have the same parameters, we can include their arguments in the same columns in the pseudo-DM.
# define turning angle pseudo-DM
# b1 b2 b3
angleDM <- matrix(c(0, 0, 0,# mean resident
"I(land_dir + 3.14/2)",0, 0,# mean travel_L
"I(land_dir - 3.14/2)",0, 0,# mean travel_R
0, 1, 0, # k resident
0, 0, 1, # k travel_L
0, 0, 1),# k travel_R
byrow = T, nrow = nbState*2,
dimnames = list(c("mean_R","mean_TL","mean_TR",
"k_R","k_TL","k_TR"),
c("mean_T:land","k_R","k_T")))In the above pseudo-DM, the coefficients mean_T:land and k_T will be shared between travel_L and travel_R as the rows are identical.
To remove additional parameters, we will also specify the pseudo-DM for the step length with shared parameters for the travel states.
# define turning angle pseudo-DM
# b1 b2 b3 b4
stepDM <- matrix(c(1, 0, 0, 0,# mean resident
0, 1, 0, 0,# mean travel_L
0, 1, 0, 0,# mean travel_R
0, 0, 1, 0, # sd resident
0, 0, 0, 1, # sd travel_L
0, 0, 0, 1),# sd travel_R
byrow = T, nrow = nbState*2,
dimnames = list(c("mean_R","mean_TL","mean_TR",
"sd_R","sd_TL","sd_TR"),
c("mean_R","mean_T", "sd_R","sd_T")))
# combine DMs
DM <- list(step = stepDM, angle = angleDM)We can also share the transition probabilities to and from the travel states using the beta constraints (betaCons) argument in the fitHMM function. betaCons is a matrix representing the off-diagonal elements of the state transition probability matrix (i.e., not including the probabilities of remaining in the same state). The values of the matrix must be integers identifying any equality constraints (identical integers), wherein the integer must represent the lowest index of constrained values (see betaCons details in help("fitHMM")).
# To: R,TL,TR
betaCons <- matrix(c( 1, 1, # from R
3, 4, # from TL
3, 4 ), # from TR
nrow = 1, byrow = T)In the above code, we left the diagonal values (remaining in the same state) empty for visual clarity. The two 1s convey that the transition from resident to either travel_L or travel_R is the same. No index of “2” (from resident to travel_R) is used as it is constrained to index “1”. The 3s convey that the probabilities of transitioning from travel_L to resident or travel_R to resident are the same. Last, the 4s convey that the probability of transitioning from travel_L to travel_R is the same as from travel_R to travel_L.
Now, we can specify the starting parameters and fit the HMM.
set.seed(1)
# define starting values
mu0 <- log(c(200, 500)) # SL mean (500 shared by TL and TR)
sigma0 <- log(c(150, 300)) # SL sd (300 shared by TL and TR)
angle0 <- 5 # TA mean
kappa0 <- log(c(0.1, 1)) # TA kappa (1 shared by TL and TR)
Par0 <- list(step = c(mu0, sigma0), angle = c(angle0,kappa0))
# Fit HMM
HMM_AngleLand <- fitHMM(prep_gps_crw_wgs84, nbState = nbState,
dist = dist, Par0 = Par0, DM = DM,
stateNames = stateNames,
estAngleMean = list(angle = T),
circularAngleMean = list(angle = T),
betaCons = betaCons)
# plot
plot(HMM_AngleLand, ask = FALSE, plotCI = T, plotTracks = F)## Decoding state sequence... DONE
# show parameter estimates and confidence intervals
# get estimate and CI
HMM_AngleLand$CIbeta$angle[c("est", "lower", "upper")] %>%
lapply(as.vector) %>% # convert to vectors
do.call(cbind, .) %>% # cbind into data.frame
# add beta names
data.frame(names = colnames(HMM_AngleLand$mle$angle), .) %>%
kable(digits = 3) # output table| names | est | lower | upper |
|---|---|---|---|
| mean_T:land | 0.162 | 0.129 | 0.195 |
| k_R | 0.819 | 0.746 | 0.892 |
| k_T | 3.077 | 2.958 | 3.195 |
Given the confidence intervals to not include 0, it appears that at least some of narwhal orientation during travel is governed by the direction of the coastline. We can map these states with the coastline using tmap. For clarity, we will only plot one individual.
# identify most likely state using the Viterbi algorithm
prep_gps_crw_wgs84$state <-
viterbi(HMM_AngleLand)
# convert to SF object
prep_gps_crw_sf_sub <- prep_gps_crw_wgs84 %>%
filter(ID == "T172066") %>%
st_as_sf(coords = c("x", "y")) %>% # converts to an sf object
st_set_crs(2962)
# plot with tmap
land_sf <- st_read("data/land.shp") # read land as sf object## Reading layer `land' from data source
## `G:\My Drive\Work\Education\DFO_HMM_2022\Day2\data\land.shp'
## using driver `ESRI Shapefile'
## Simple feature collection with 1 feature and 10 fields
## Geometry type: MULTIPOLYGON
## Dimension: XY
## Bounding box: xmin: -320000 ymin: 8140000 xmax: -150000 ymax: 8250000
## Projected CRS: NAD83(CSRS) / UTM zone 21Ntm_shape(land_sf) +
tm_polygons() +
tm_shape(prep_gps_crw_sf_sub) +
tm_dots(col = "state", palette = c("brown", "dodgerblue", "darkblue"),
size = 0.1)
With clever definition of the direction to shore, it should be possible to define travel into fjords and travel out of fjords as two distinct states, which may be ecologically interesting.
7.2.3 BCRW - bias toward an unknown angle
If the animal is presumed to move at some angle other than \(0^\circ\), \(180^\circ\), or \(\pm90^\circ\) that is not known a priori, we can allow the model to estimate the angle of bias by defining animal movement as a trade-off between directional persistence, bias in the same direction as the stimulus, and bias perpendicular to the stimulus. For example, if we thought narwhals moved at some unknown angle relative to the direction of the tides, the pseudo-DM could be formatted as follows:
# define turning angle pseudo-DM
# b1 b2 b3 b4
angleDM <- matrix(c(0, 0, 0, 0,# mean resident
"tide", "I(tide + 3.14/2)",0, 0,# mean travel_L
"tide", "I(tide - 3.14/2)",0, 0,# mean travel_R
0, 0, 1, 0, # k resident
0, 0, 0, 1, # k travel_L
0, 0, 0, 1),# k travel_R
byrow = T, nrow = nbState*2,
dimnames = list(c("mean_R","mean_TL","mean_TR",
"k_R","k_TL","k_TR"),
c("mean_T:tide","mean_T:tide90",
"k_R","k_T")))Once the model is fit and the beta coefficients are estimated, the angle of attraction can be calculated using atan2(b2, b1). See Togunov et al. (2021) and the accompanying tutorial for details on fitting a menotactic BCRW using momentuHMM.